IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
First order shear deformation (FSDT) theory for laminated composite beams is used to study free vibration of
laminated composite beams, and finite element method (FEM) is employed to obtain numerical solution of the
governing differential equations. Free vibration analysis of laminated beams with rectangular cross – section for
various combinations of end conditions is studied. To verify the accuracy of the present method, the frequency
parameters are evaluated and compared with previous work available in the literature. The good agreement with
other available data demonstrates the capability and reliability of the finite element method and the adopted beam
model used.
Contradictory of the Laplacian Smoothing Transform and Linear Discriminant An...TELKOMNIKA JOURNAL
Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES)
Ad hoc & sensor networks, Adaptive applications, Aeronautical Engineering, Aerospace Engineering
Agricultural Engineering, AI and Image Recognition, Allied engineering materials, Applied mechanics,
Architecture & Planning, Artificial intelligence, Audio Engineering, Automation and Mobile Robots
Automotive Engineering….
Numerical approach of riemann-liouville fractional derivative operatorIJECEIAES
This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.
First order shear deformation (FSDT) theory for laminated composite beams is used to study free vibration of
laminated composite beams, and finite element method (FEM) is employed to obtain numerical solution of the
governing differential equations. Free vibration analysis of laminated beams with rectangular cross – section for
various combinations of end conditions is studied. To verify the accuracy of the present method, the frequency
parameters are evaluated and compared with previous work available in the literature. The good agreement with
other available data demonstrates the capability and reliability of the finite element method and the adopted beam
model used.
Contradictory of the Laplacian Smoothing Transform and Linear Discriminant An...TELKOMNIKA JOURNAL
Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES)
Ad hoc & sensor networks, Adaptive applications, Aeronautical Engineering, Aerospace Engineering
Agricultural Engineering, AI and Image Recognition, Allied engineering materials, Applied mechanics,
Architecture & Planning, Artificial intelligence, Audio Engineering, Automation and Mobile Robots
Automotive Engineering….
Numerical approach of riemann-liouville fractional derivative operatorIJECEIAES
This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
A Computationally Efficient Algorithm to Solve Generalized Method of Moments ...Waqas Tariq
Generalized method of moment estimating function enables one to estimate regression parameters consistently and efficiently. However, it involves one major computational problem: in complex data settings, solving generalized method of moments estimating function via Newton-Raphson technique gives rise often to non-invertible Jacobian matrices. Thus, parameter estimation becomes unreliable and computationally inefficient. To overcome this problem, we propose to use secant method based on vector divisions instead of the usual Newton-Raphson technique to estimate the regression parameters. This new method of estimation demonstrates a decrease in the number of non-convergence iterations as compared to the Newton-Raphson technique and provides reliable estimates.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Numerical Solutions of Second Order Boundary Value Problems by Galerkin Resid...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
A Computationally Efficient Algorithm to Solve Generalized Method of Moments ...Waqas Tariq
Generalized method of moment estimating function enables one to estimate regression parameters consistently and efficiently. However, it involves one major computational problem: in complex data settings, solving generalized method of moments estimating function via Newton-Raphson technique gives rise often to non-invertible Jacobian matrices. Thus, parameter estimation becomes unreliable and computationally inefficient. To overcome this problem, we propose to use secant method based on vector divisions instead of the usual Newton-Raphson technique to estimate the regression parameters. This new method of estimation demonstrates a decrease in the number of non-convergence iterations as compared to the Newton-Raphson technique and provides reliable estimates.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Numerical Solutions of Second Order Boundary Value Problems by Galerkin Resid...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Static and Dynamic Reanalysis of Tapered BeamIJERA Editor
Beams are one of the common types of structural components and they are fundamentally categorized as
uniform and non-uniform beams. The non-uniform beams has the benefit of better distribution of strength and
mass than uniform beam. And non-uniform beams can meet exceptional functional needs in
aeronautics,robotics,architecture and other unconventional engineering applications. Designing of these
structures is necessary to resist dynamic forces such as earthquakes and wind.
The present paper focuses on static and dynamic reanalysis of a tapered cantilever beam structure using
multipolynomial regression method. The method deals with the characteristics of frequency of a vibrating
system and the procedures that are available for the modification of physical parameters of vibrating system.
The method is applied on a tapered cantilever beam for approximate structural static and dynamic reanalysis.
Results obtained from the assumed conditions of the problem indicate the high quality approximation of stresses
and natural frequencies using ANSYS and Regression method.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
Engineering Research Publication
Best International Journals, High Impact Journals,
International Journal of Engineering & Technical Research
ISSN : 2321-0869 (O) 2454-4698 (P)
www.erpublication.org
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Structural Dynamic Reanalysis of Beam Elements Using Regression MethodIOSR Journals
This paper concerns with the reanalysis of Structural modification of a beam element based on
natural frequencies using polynomial regression method. This method deals with the characteristics of
frequency of a vibrating system and the procedures that are available for the modification of physical
parameters of vibrating structural system. The method is applied on a simple cantilever beam structure and Tstructure
for approximate structural dynamic reanalysis. Results obtained from the assumed conditions of the
problem indicates the high quality approximation of natural frequencies using finite element method and
regression method.
Topic: Regression
Student Name: Nayab
Class: B.Ed. 2.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
Implementation Of Geometrical Nonlinearity in FEASTSMTiosrjce
Analysis of the structures used in aerospace applications is done using finite element
method. These structures may face unexpected loads because of variable environmental situations.
These loads could lead to large deformation and inelastic manner. The aim of this research is to
formulate the finite elements considering the effect of large deformation and strain. Here total
Lagrangian method is used to consider the effect of large deformation. After deriving required
relations, implementation of formulated equation is done in FEASTSMT(Finite Element Analysis of
Structures - Substructured and Multi-Threading). .Newton-Raphson method was utilized to solve
nonlinear finite element equations. The validation is carried out with the results obtained from the
Marc Software.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Applied Numerical Methods Curve Fitting: Least Squares Regression, InterpolationBrian Erandio
Correction with the misspelled langrange.
and credits to the owners of the pictures (Fantasmagoria01, eugene-kukulka, vooga, and etc.) . I do not own all of the pictures used as background sorry to those who aren't tagged.
The presentation contains topics from Applied Numerical Methods with MATHLAB for Engineers and Scientist 6th and International Edition.
A Linear Assignment Formulation Of The Multiattribute Decision Problem
Mx2421262131
1. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2126-2131
Dynamic Reanalysis of Beams Using Polynomial Regression
Method
*
B. Rama Sanjeeva Sresta, **Dr. Y. V. Mohan Reddy
*
(Department of Mechanical Engineering, G Pulla Reddy Engineering College, Kurnool, A.P., India,)
**
(Department of Mechanical Engineering, G Pulla Reddy Engineering College, Kurnool, A.P., India,)
ABSTRACT:
The paper focuses on dynamic reanalysis the complete set of modified simultaneous equations.
of simple beam structures using a polynomial The solution procedures usually use the original
regression method. The method deals with the response of the structure.
stiffness and mass matrices of structures and can
be used with a general finite element system. This SOLUTION APPROACH- FINITE
method is applied to approximate dynamic ELEMENT METHOD
reanalysis of cantilever simple beam structure and Initially the beam is divided into smaller
T-structure. Preliminary results for these example sections using successive levels of division. Analysis
problems indicate the high quality approximation of each section is performed separately. Using the
of natural frequencies can be obtained. The final finite element technique, the dynamic analysis of
results from regression method and Finite element beam structure is modeled.
method are compared. [K-λM] [X]=0 ----------------- (1)
Where k, m are the stiffness and mass matrix
Keywords: mass matrix, stiffness matrix, respectively.
natural frequency, dynamic reanalysis, The dynamic behavior of a damped structure
polynomial regression. [4] which is assumed to linear and discretized for n
degrees of freedom can be described by the equation
INTRODUCTION: of motion.
Reanalysis methods are intended to analyze M +C +Kx=f-------------(2)
efficiently new designs using information from Where M, C = αM+βK, and K are mass,
previous ones. One of the many advantages of the damping and stiffness matrices, , and X are
substructure technique is the possibility of repeating acceleration, velocity, displacement vectors of the
the analysis for one or more of the substructures structural points and “f” is force vector. Undamped
making use of the work done on the others. This homogeneous equation M +Kx=0. Provides the
represents a significant saving of time when
Eigen value problem (k-λm) = 0.
modifications once are required. Modification is
Solution of above equation yields the matrices Eigen
invariably required in iterative processes for optimum
design never the less, in the case of large structures values λ and Eigen vectors
the expenses are still too high.[1]
Therefore, development of techniques which λ = , = [ 1, 2….. n]
are themselves based on previous analysis, and which
obtained the condensed matrices of the substructures The eigenvector satisfy the orthonormal
under modification, with little extra calculation time, conditions M =I, K =λ, C = αI+βλ=ξ,
can be very useful. “General Reanalysis Techniques” Using the transformation X = q in the equation of
are very useful in solving medium size problems and motion, and premultiplying by one obtains,
are totally essential in the design of large structures.
M + C + K q= f -------------(3)
Some steps in a dynamic condensation process are
particularly characterized by their computational It is important note, that the matrices,
effort, as for instance: = M , = C , K are not
Stiffness matrix factorization usually diagonalised by the eigenvectors of the
Resolution of certain systems of linear equation original structure [3] Given an initial geometry and
Resolution of an eigen problem to obtain the assuming a change ΔY in the design variables, the
normal vibration modes. modified design is given by
Reanalysis methods [2] are intended to Y = +ΔY. ------------------ (4)
analyze efficiently structures that are modified due to The geometric variables Y usually represent
changes in the design. The object is to evaluate the coordinates of joints, but other choice for these
structural response for such changes without solving
2126 | P a g e
2. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2126-2131
variables is sometimes preferred. The displacement The parameter, α, is a constant, often called
analysis equations for the initial design are the “intercept” while b is referred to as a regression
r = R. coefficient that corresponds to the “slope” of the line.
where = stiffness matrix corresponding to the The additional parameter ε accounts for the type of
error that is due to random variation caused by
design , R= load vector whose elements are usually experimental imprecision. The regression procedure
assumed to be independent of the design variables assumes that the scatter of the data points about the
and r= nodal displacements computed at . The best-fit straight line reflects the effects of the error
stiffness matrix and mass matrix of a typical plane term,[12-15] and it is also implicitly assumed that ε
truss element are follows a Gaussian distribution with a mean of 0.
Now, however, we will assume that the error is
K= and Gaussian Figure 2 illustrates the output of the linear
model with the inclusion of the error term.
3.1.3 Multiple Linear Regressions
M= The straight line equation is the simplest
form of the linear regression given as
Y=α+βX+ε
Where α+βX represents the deterministic part and ε is
„A‟ is the cross sectional area, ‟l‟ is member length, the stochastic component of the model.
of the beam „ „ is density of the beam.
The simple linear population model equation
REANALYSIS OF REGRESSION indicating the deterministic component of the model
METHOD that is precisely determined by the parameters α and
The statistical determination of the β, and the stochastic component of the model, ε that
relationship between two or more dependent represents the contribution of random error to each
variables has been referred to as a correlation determined value of Y. It only includes one
analysis, [6]whereas the determination of the independent variable. When the relationship of
relationship between dependent and independent interest can be described in terms of more than one
variables has come to be known as a regression independent variable, the regression is then defined
analysis. as “multiple linear regression.” The general form of
the linear regression model may thus be written as:
1.1 Regression Y = +…….+ + ε. ---------------
The actual term “regression” is derived from (6)
the Latin word “regredi,” and means “to go back to” Where, Y is the dependent variable, and X1,
or “to retreat.” Thus, the term has come to be X2 … Xi are the (multiple) independent variables.
associated with those instances where one “retreats” Multiple linear regression models also encompass
or “resorts” to approximating a response variable polynomial functions:
with an estimated variable based on a functional Y= +……. + +, ------------- (7)
relationship between the estimated variable and one The equation for a straight line is a first-order
or more input variables. In regression analysis, the polynomial. The quadratic equation,
input (independent) variables can also be referred to Y= ------------------ (8)
as “regressor” or “predictor” variables.
is a second-order polynomial whereas the cubic
equation,
3.1.1 Linear Regression
Linear regression involves specification of a Y= -------------- (9)
linear relationship between the dependent variable(s) is a third-order polynomial.
and certain properties of the system under Taking first derivatives with respect to each of the
investigation. Linear regression deals with some parameters yields:
curves as well as straight lines. = 1, = X, = ---------------- (10)
The model is linear because the first
derivatives do not include the parameters. As a
3.1.2 Ordinary Linear Regression consequence, taking the second (or higher) order
The simplest general model for a straight derivative of a linear function with respect to its
line includes a parameter that allows for inexact fits: parameters will always yield a value of zero. Thus, if
an “error parameter” which we will denote as . the independent variables and all but one parameter
Thus we have the formula: are held constant, the relationship between the
dependent variable and the remaining parameter will
Y = α +β X + ----------------- (5)
always be linear. It is important to note that linear
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3. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2126-2131
regression does not actually test whether the data user. Figure (2) shows an example of creating a finite
sampled from the population follow a linear element for a cantilever beam.
relationship. It assumes linearity and attempts to find
the best-fit straight line relationship based on the data
sample. The dashed line shown in the figure (1) is the
deterministic component, whereas the points
represent the effect of random error.
Figure 2: Descretized Element
The polynomial equation for regression method,
+ + .
These 3 values for both case studies
Figure 1: A linear model that incorporates a Young‟s modulus (E) 0.207× N/
stochastic (random error) component. Density (ρ) 7806 Kg/
Cross section of area (A) 0.029×0.029
3.1.4 Assumptions of Standard Regression
Analyses
1.2 Case Study 1
The subjects are randomly selected from a larger The Cantilever Beam of 1m length, shown
population. The same caveats apply here as with in figure () is divided into 4 elements equally element
correlation analyses. The observations are Stiffness Matrix and Mass Matrix are extracted.
independent. The variability of values around the Natural frequencies of the cantilever beam at each
line is Gaussian. node are found from MATLAB program by
X and Y are not interchangeable. Regression considering two situations-
models used in the vast majority of cases attempt a) width alone is increased by 5% and
to predict the dependent variable, Y, from the b) width and depth of the beam are increased by 5%
independent variable, X and assume that the each.
error in X is negligible. In special cases where Reanalysis of the beam is done by
this is not the case, extensions of the standard Polynomial regression and the percentage errors are
regression techniques have been developed to listed in the table.
account for non negligible error in X.
The relationship between X and Y is of the
correct form, i.e., the expectation function (linear
or nonlinear model) is appropriate to the data
being fitted.
There are enough data points to provide a good
sampling of the random error associated with the
Experimental observations. In general, the
minimum number of independent points can be
no less than the number of parameters being Figure 3: Cantilever beam with nodes and
estimated, and should ideally be significantly elements
higher.
First natural frequencies of cantilever beam
NUMERICAL EXAMPLES by increasing depth of the beam by 5%. The
The polynomial regression method is polynomial regression equation is given by
applied to a simple beam structures. In finite element + +
method, Discretization means dividing the body into Fitting target of lowest sum of squared absolute error
an equivalent system of finite elements with = 8.7272727289506574 ,
associated nodes. The element must be made small -3.6333 , = -
enough to view and give usable results and to be
1.053659
large enough to reduce computational efforts. Small
elements are generally desirable where the results are = 7.76275 , = -
changing rapidly such as where the changes in 3.05561 ,
geometry occur. Large elements can be used where = 8.07983 C6 = 2.2512E+01
the results are relatively constant. The discretized
body or mesh is often created with mesh generation First natural frequencies of cantilever beam
program or preprocessor programs available to the by increasing width and depth of the beam by 5%.
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4. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2126-2131
Fitting target of lowest sum of squared absolute error
= 1.8470358974358964 ,
-6.30489 , = 4.2 Case Study 2
4.028518 , The dimensions of a T-structure are given in
figure (4). The Cantilever Beam is divided into 6
= 4.028518 , = -
elements. Then Element Stiffness Matrix, mass
1.0347691 , Matrix and natural frequencies are determined using
= -1.0347691 , = - MATLAB. Polynomial regression method is applied
1.0347691 , to this structure. The results from polynomial
regression and FEM are compared for closeness.
Table 1: Increasing the Depth of the Beam
Natural Natural
Widt Frequenc Frequency %Erro
Height
h y Regression(H r
Fem(Hz) z)
0.029 0.029 22.53 22.52 -0.044
0.0304
0.029 23.65 23.64 -0.042
5
0.029 0.0319 24.78 24.69 -0.363
0.0333
0.029 25.91 25.88 -0.115
5
0.029 0.0348 27.03 27.01 -0.074 Figure 4: T-Structure with nodes and elements
0.0362
0.029 28.16 28.13 -0.106
5 The results are as follows: First natural
0.029 0.0377 29.29 29.27 -0.020 frequencies of cantilever beam by increasing depth of
0.0391 the beam by 5%.
0.029 30.41 30.40 -0.033
5 Fitting target of lowest sum of squared absolute error
0.029 0.0406 31.54 31.53 -0.032 equal to 1.0240640782836770 ,
0.0420 -8.043993 , = -
0.029 32.67 32.66 -0.03
5 1.35761 ,
0.029 0.0435 33.79 33.76 -0.089
= 1.53493 , = -
6.65185
Table 2: Increasing Width and Depth of the Beam
Width Height Natural Natural %Erro = -3.99731 , =
Frequen Frequency r 4.446583 .
cy Regression(
Fem(Hz Hz) Natural frequencies of cantilever beam by
) increasing width and depth of the beam by 5%,
0.029 0.029 22.47 22.47 0 Fitting target of lowest sum of squared absolute error
0.0304 0.0304 23.60 23.61 0.0423 equal to 7.2417839254080818 ,
5 5 7
0.0319 0.0319 24.72 24.75 0.1213 -3.483496 =
5
0.0333 0.0333 26.08 26.89 3.1058 7.71712 ,
5 5 2
0.0348 0.0348 26.97 27.03 0.2224 = 7.71712 , =-
6
0.0362 0.0362 28.09 28.16 0.2491
1.8478
5 5 9
0.0377 0.0377 29.22 29.59 1.2662
5 = -1.8478 =-
0.0391 0.0391 30.34 30.38 0.1318
5 5 3 1.8478
0.0406 0.0406 31.56 31.90 1.0773
1
0.0420 0.0420 32.59 32.70 0.3375
5 5 2
0.0435 0.0435 33.80 33.83 0.0887
5
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5. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2126-2131
Table 3:Increasing Depth of the Beam CONCLUSION
Natural Natural From this work the following conclusions
Frequen Frequenc are drawn. The FEM method is applied for dynamic
Widt
Height cy y %Error analysis of cantilever beam and T-structure using the
h
Fem Regressi MAT lab. Natural frequencies are obtained for
(Hz) on (Hz) cantilever and T-structure beams using FEM.the
0.02 4.4975× polynomial regression method is used for obtaining
0.029 444.69 444.71 natural frequencies of cantilever beam and T-
9
0.02 0.0304 4.2834× structure by varying width and depth for dynamic
466.91 466.93 reanalysis.
9 5
2.0443× The results obtained from reanalysis using
0.02
0.0319 489.16 489.17 regression method are close to results obtained using
9
FEM. The minimum and maximum errors in
0.02 0.0333 1.9550× regression method when compared with the results
511.49 511.50
9 5 obtained by FEM are
0.02 3.7479×
0.0348 533.63 533.65
9 Minimum Maximum
0.02 0.0362 1.7990× -0.36319 (with -0.02000
555.85 555.86 Cantilever increasing width) 3.10582
9 5
0.02 3.4600× beam 0 (with increasing
0.0377 578.02 578.04 width and depth)
9
1.6657× 0(with increasing
0.02 0.0391 4.4975×
600.33 600.34 width)
9 5 T-structure 0.302235
0(with increasing
0.02 width and depth)
0.0406 622.57 622.57 0
9
0.02 0.0420 3.1019×
644.76 644.78 REFERENCES:
9 5 [1]. M.M.Segura and J.T.celigilete, “ a new
0.02 2.0084× dynamic reanalysis technique based on
0.0435 667.02 667.04
9 model synthesis” “Centro de studios e
investigaciones tecnicas”, de
Table 4: Increasing Width and Depth of the guipuzcoa(CEIT) apartado 1555, 20009 san
Beam(T-Structure) Sebastian, spain (1994).
Width Heigh Natural Natural %Error [2]. Z. Xie, W.S. Shepard Jr,” Development of a
t Frequen Frequency single-layer finite element and a simplified
cy Regression( finite element modelling approach for
Fem(Hz Hz) constrained layer damped structures”, Finite
) Elements in Analysis and Design 45 (2009)
0.029 0.029 444.36 444.36 0 pp530–537.
0.030 0.030 465.79 466.43 0.1374 [3]. Uri kirsch , Michaelbogomolni , “analytical
45 45 00 modification of structural natural
0.031 0.031 487.97 488.81 0.1721 frequencies”Department of civil and
9 9 41 environmental engineering ,Technion- Israel
0.033 0.033 514.76 515.18 0.1787 institute of technology, Haifa 32000, Israel
35 35 2 [2006].
0.034 0.034 532.30 533.55 0.2347 [4]. Uri kirsch, G.Tolendano,”Finite Elements in
8 8 41 Analysis and Design “Department of Civil
0.036 0.036 554.51 555.93 0.2560 and Environmental engineering , Technion
25 25 81 Israel institute of technology, Haifa 32000,
0.037 0.037 576.72 578.30 0.2739 Israel.
7 7 63 [5]. M. Nad‟a, “structural dynamic modification
0.039 0.039 598.87 600.68 0.3022 of vibrating system”* a Faculty of Materials
15 15 35 Science and Technology, STU in Bratislava,
0.040 0.040 629.67 630.65 0.1556 Paulínska 16, 917 24 Trnava, Slovak
6 6 35 Republic.
0.042 0.042 643.26 645.43 0.1818 [6]. Tirupathi R.Chandrupatla , “Ashok
05 05 86 D.Belegundu ,Rowan University
Glassboro”, new jersey and the
0.043 0.043 667.06 667.79 0.1094
Pennsylvania state university, university
5 5 35
park, Pennsylvania.
2130 | P a g e
6. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2126-2131
[7]. Arthur Christopoulos and Michael J. Lew [11]. Hill, A.V., “The combinations of
Critical Reviews in Biochemistry and haemoglobin with oxygen and with carbon
Molecular Biology, 35(5):359–391 (2000). monoxide.” I, Biochem. J., 7, 471– 80,1913.
[8]. Hill, A.V., The combinations of [12]. Motulsky, H.J., “Intuitive Biostatistics”,
haemoglobinwith oxygen and with carbon Oxford University Press, New York, 1995.
monoxide. I, Biochem. J., 7, 471 80, 1913. [13]. Motulsky, H.J., “Analyzing Data with
[9]. Wells, J.W., “Analysis and interpretation of GraphPad Prism”, GraphPad Software Inc.,
binding at equilibrium, in Receptor-Ligand San Diego,CA, 1999.
Interactions”: A Practical Approach, E.C. [14]. Ludbrook, J., “Comparing methods of
Hulme, Ed. Oxford University Press, measurements”, Clin. Exp. Pharmacol.
Oxford, 1992, 289–395. Physiol.,24(2), 193–203, 1997.
[10]. Hill, A.V., “The possible effects of the [15]. Bates, D.M. and Watts, D.G., “Nonlinear
aggregation of the molecules of Regression Analysis and Its Applications”,
haemoglobin on its dissociation curves”, J Wiley and Sons, New York, 1988.
. Physiol. , 40, iv–vii,1910.
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